Inventory resonances in multi-echelon supply chains

摘要

We consider multi-echelon supply chains, typically consisting of retailer, manufacturer, supplier, etc., in which each actor chooses its own inventory replenishment strategy, and is subject to its own replenishment delay. We model the chain with a system of linear differential delay equations. In matrix form the system of equations is triangular and we derive an exact, algorithmic solution, which allows us to determine the inventory behaviour over time for each echelon. The solution is characterised by a set of Lambert modes, obtained from the solution to a set of transcendental equations. For a complete set of solutions the modes must be independent. When the modes are not independent, we find that additional solutions exist, and these exhibit a transient surge in inventory. In analogy with ordinary differential equations, we refer to this as a 'resonance' in the inventory. We discuss the practical import of this theoretical prediction.